Measure-theoretic aspects of Convex bodies

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。提出的研究对象是研究高维凸体的质量分布或更一般的对数-凹概率测度。最近，概率论的几个经典结果已经扩展到这些措施的更广泛的设置。这些新的结果似乎超出了基于独立性的经典概率推理的范围，后者被凸性的几何概念所取代。PI打算进一步研究高维测度的几何参数，特别是与这些测度相关的广义质心体的几何参数。利用这种方法，并应用Banach空间的局部理论、经典凸性、概率论和信息论的技术，PI希望解决与对数凹测度相关的各种开放问题，如小球概率估计和超平面猜想，以及凸体理论中的各种猜想，如熵数的正则性和最小平均宽度的最优界。在这个提议的研究中有一个普遍的原则：高维系统倾向于围绕典型形式聚集。这是影响在概率论、组合学、统计物理学和复杂性中出现的复杂系统研究的核心事实。当我们把高维凸体看作概率空间时，我们期望它的意想不到的规律性为我们对这一一般原理的理解增加一个新的组成部分。随机多面体、随机矩阵和随机算法（分别来自组合学、数学物理和信息学的主题）的应用已经被发现，这表明未来应该期待更多的应用。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The object of the proposed research is the study of the distribution of mass of high dimensional convex bodies or more general of log-concave probability measures. Recently, several classical results from probability theory have been extended to the broader setting of these measures. These new results appear to be out of reach of the classical probabilistic reasoning based on independence which is replaced by the geometric notion of convexity. The PI intends to further investigate the geometric parameters of high-dimensional measures and in particular the geometry of generalized centroid bodies associated to these measures. With this approach and applying techniques from local theory of Banach spaces, classical convexity, probability and information theory, the PI wishes to attack various open problems related to log-concave measures such as small ball probability estimates and the hyperplane conjecture, as well as, various conjectures in the theory of convex bodies such as the regularity of the entropy numbers and the optimal bound of the minimal mean width. There is a general principle that underlies the research in this proposal: the tendency of high dimensional systems to congregate around typical forms. This is a central fact that influences the study of complex systems which appear in probability, combinatorics, statistical physics and complexity. The unexpected regularity of high dimensional convex bodies when viewed as probability spaces is expected to add a new component to our understanding of this general principle. Applications to random polytopes, random matrices, and random algorithms (topics from combinatorics, mathematical physics and informatics respectively) have already been discovered, indicating thatmore should be expected in the future.